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i'm having trouble getting this one started please.

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\frac{\sec x + \csc x}{1 + \tan x} \qquad \sin x$$

I have tried converting to

$$\frac{\dfrac{1}{\cos x} + \dfrac{1}{\sin x}}{1 + \dfrac{\sin x}{\cos x}}$$

Then $$\frac{ 1 + \dfrac{\cos x}{\sin x}}{\cos x + \sin x}$$

But if I progress this further I cannot seem to yield a result and I'm not sure if I'm heading in the right direction witht this.

The answer is apparently, $\dfrac{1}{\sin x}$

Blue
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Bucephalus
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    $ \displaystyle \frac{ 1 + \frac{\cos x}{\sin x}} {\cos x + \sin x} = \frac{ 1 + \frac{\cos x}{\sin x}} {\sin x \cdot (1 + \frac{\cos x}{\sin x})}$ – Math Lover Mar 12 '22 at 07:52
  • Oh yeah, that's really good. Do you want to put that in an answer and I will vote it up? @MathLover – Bucephalus Mar 12 '22 at 08:03
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    @Bucephalus As a general tip, if you know what the expression will simplify to, then try to force the expression to make it look like the desired result, e.g. in this case, that amounts to factoring out a $\sin x$ on the bottom from the very start – Golden_Ratio Mar 12 '22 at 08:09
  • Oh yeah, that makes sense @Golden_Ratio . Thanks for the tip. – Bucephalus Mar 12 '22 at 08:16
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    @Bucephalus ok. I just added an answer. – Math Lover Mar 12 '22 at 08:17
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    Alternatively, $\frac{\sin x + \cos x}{\sin x \cos x + \sin^2 x} = \frac{\sin x + \cos x}{\sin x (\cos x + \sin x)} = \frac{1}{\sin x}.$ – Toby Mak Mar 12 '22 at 08:42
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    Thank you @TobyMak – Bucephalus Mar 12 '22 at 09:40

2 Answers2

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We can rewrite numerator as,

$ \displaystyle \csc x + \sec x = \csc x \cdot \left(1 + \frac{\sec x}{\csc x}\right)$

$ \displaystyle~~~~ = \csc x \left(1 + \frac{\sin x}{\cos x}\right) = \csc x \left(1 + \tan x\right)$

So, $\displaystyle \frac {\csc x + \sec x}{1 + \tan x} = \frac{1}{\sin x}$

Math Lover
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By definition, $$ \begin{aligned} \frac{\frac{1}{\cos x}+\frac{1}{\sin x}}{1+\frac{\sin x}{\cos x}} &=\frac{\frac{\sin x+\cos x}{\cos x \sin x}}{\frac{\cos x+\sin x}{\cos x}} \\ &=\frac{1}{\sin x} \end{aligned} $$

Lai
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